theory IntEx
imports QuotMain
begin

fun
  intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" (infix "\<approx>" 50)
where
  "intrel (x, y) (u, v) = (x + v = u + y)"

quotient my_int = "nat \<times> nat" / intrel
  apply(unfold equivp_def)
  apply(auto simp add: mem_def expand_fun_eq)
  done

thm quotient_thm

thm my_int_equivp

print_theorems
print_quotients

quotient_def 
  ZERO::"my_int"
where
  "ZERO \<equiv> (0::nat, 0::nat)"

ML {* print_qconstinfo @{context} *}

term ZERO
thm ZERO_def

ML {* prop_of @{thm ZERO_def} *}

ML {* separate *}

quotient_def 
  ONE::"my_int"
where
  "ONE \<equiv> (1::nat, 0::nat)"

ML {* print_qconstinfo @{context} *}

term ONE
thm ONE_def

fun
  my_plus :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
where
  "my_plus (x, y) (u, v) = (x + u, y + v)"

quotient_def 
  PLUS::"my_int \<Rightarrow> my_int \<Rightarrow> my_int"
where
  "PLUS \<equiv> my_plus"

term my_plus
term PLUS
thm PLUS_def

fun
  my_neg :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
where
  "my_neg (x, y) = (y, x)"

quotient_def 
  NEG::"my_int \<Rightarrow> my_int"
where
  "NEG \<equiv> my_neg"

term NEG
thm NEG_def

definition
  MINUS :: "my_int \<Rightarrow> my_int \<Rightarrow> my_int"
where
  "MINUS z w = PLUS z (NEG w)"

fun
  my_mult :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
where
  "my_mult (x, y) (u, v) = (x*u + y*v, x*v + y*u)"

quotient_def 
  MULT::"my_int \<Rightarrow> my_int \<Rightarrow> my_int"
where
  "MULT \<equiv> my_mult"

term MULT
thm MULT_def

(* NOT SURE WETHER THIS DEFINITION IS CORRECT *)
fun
  my_le :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"
where
  "my_le (x, y) (u, v) = (x+v \<le> u+y)"

quotient_def 
  LE :: "my_int \<Rightarrow> my_int \<Rightarrow> bool"
where
  "LE \<equiv> my_le"

term LE
thm LE_def


definition
  LESS :: "my_int \<Rightarrow> my_int \<Rightarrow> bool"
where
  "LESS z w = (LE z w \<and> z \<noteq> w)"

term LESS
thm LESS_def

definition
  ABS :: "my_int \<Rightarrow> my_int"
where
  "ABS i = (if (LESS i ZERO) then (NEG i) else i)"

definition
  SIGN :: "my_int \<Rightarrow> my_int"
where
 "SIGN i = (if i = ZERO then ZERO else if (LESS ZERO i) then ONE else (NEG ONE))"

ML {* print_qconstinfo @{context} *}

lemma plus_sym_pre:
  shows "my_plus a b \<approx> my_plus b a"
  apply(cases a)
  apply(cases b)
  apply(auto)
  done

lemma list_equiv_rsp[quotient_rsp]:
  shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>"
by auto

lemma ho_plus_rsp[quotient_rsp]:
  shows "(intrel ===> intrel ===> intrel) my_plus my_plus"
by (simp)

ML {* val qty = @{typ "my_int"} *}
ML {* val ty_name = "my_int" *}
ML {* val rsp_thms = @{thms ho_plus_rsp} *}
ML {* val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} qty *}
ML {* val (trans2, reps_same, absrep, quot) = lookup_quot_thms @{context} "my_int"; *}

ML {* fun lift_tac_intex lthy t = lift_tac lthy t [rel_eqv] *}

ML {* fun inj_repabs_tac_intex lthy = inj_repabs_tac lthy [rel_refl] [trans2] *}
ML {* fun all_inj_repabs_tac_intex lthy = all_inj_repabs_tac lthy [rel_refl] [trans2] *}

lemma test1: "my_plus a b = my_plus a b"
apply(rule refl)
done

abbreviation 
  "Abs \<equiv> ABS_my_int"
abbreviation 
  "Rep \<equiv> REP_my_int"
abbreviation 
  "Resp \<equiv> Respects"

lemma "PLUS a b = PLUS a b"
apply(tactic {* procedure_tac @{context} @{thm test1} 1 *})
apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})
apply(tactic {* all_inj_repabs_tac_intex @{context} 1*})
apply(tactic {* clean_tac @{context} 1 *}) 
done

lemma test2: "my_plus a = my_plus a"
apply(rule refl)
done

lemma "PLUS a = PLUS a"
apply(tactic {* procedure_tac @{context} @{thm test2} 1 *})
oops

lemma test3: "my_plus = my_plus"
apply(rule refl)
done

lemma "PLUS = PLUS"
apply(tactic {* procedure_tac @{context} @{thm test3} 1 *})
oops


lemma "PLUS a b = PLUS b a"
apply(tactic {* procedure_tac @{context} @{thm plus_sym_pre} 1 *})
apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* clean_tac @{context} 1 *})
done

lemma plus_assoc_pre:
  shows "my_plus (my_plus i j) k \<approx> my_plus i (my_plus j k)"
  apply (cases i)
  apply (cases j)
  apply (cases k)
  apply (simp)
  done

lemma plus_assoc: "PLUS (PLUS x xa) xb = PLUS x (PLUS xa xb)"
apply(tactic {* procedure_tac @{context} @{thm plus_assoc_pre} 1 *})
apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})
apply(tactic {* all_inj_repabs_tac_intex @{context} 1*})
apply(tactic {* clean_tac @{context} 1 *})
done

lemma ho_tst: "foldl my_plus x [] = x"
apply simp
done

(* I believe it's true. *)
lemma foldl_rsp[quotient_rsp]:
  "((op \<approx> ===> op \<approx> ===> op \<approx>) ===> op \<approx> ===> list_rel op \<approx> ===> op \<approx>) foldl foldl"
apply (simp only: fun_rel.simps)
apply (rule allI)
apply (rule allI)
apply (rule impI)
apply (rule allI)
apply (rule allI)
apply (rule impI)
apply (rule allI)
apply (rule allI)
apply (rule impI)
apply (induct_tac xb yb rule: list_induct2) (* To finish I need to give it: arbitrary:xa ya *)
sorry

lemma "foldl PLUS x [] = x"
apply(tactic {* procedure_tac @{context} @{thm ho_tst} 1 *})
apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})
apply(tactic {* all_inj_repabs_tac_intex @{context} 1*})
apply(rule nil_rsp)
apply(tactic {* quotient_tac @{context} 1*})
apply(simp only: fun_map.simps id_simps)
apply(simp only: Quotient_abs_rep[OF fun_quotient[OF Quotient_my_int fun_quotient[OF Quotient_my_int Quotient_my_int]]])
apply(simp only: Quotient_abs_rep[OF list_quotient[OF Quotient_my_int]])
apply(simp only: foldl_prs[OF Quotient_my_int Quotient_my_int])
apply(tactic {* clean_tac @{context} 1 *})
sorry

lemma ho_tst2: "foldl my_plus x (h # t) \<approx> my_plus h (foldl my_plus x t)"
sorry

lemma "foldl PLUS x (h # t) = PLUS h (foldl PLUS x t)"
apply(tactic {* procedure_tac @{context} @{thm ho_tst2} 1 *})
apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})
apply(tactic {* all_inj_repabs_tac_intex @{context} 1*})
apply(rule cons_rsp)
apply(tactic {* quotient_tac @{context} 1 *})
apply(simp only: fun_map.simps id_simps)
apply(simp only: Quotient_abs_rep[OF fun_quotient[OF Quotient_my_int fun_quotient[OF Quotient_my_int Quotient_my_int]]])
apply(simp only: Quotient_abs_rep[OF list_quotient[OF Quotient_my_int]])
apply(simp only: foldl_prs[OF Quotient_my_int Quotient_my_int])
apply(simp only: cons_prs[OF Quotient_my_int])
apply(tactic {* clean_tac @{context} 1 *})
done